Degree of a vertex ($\mathrm{deg}(v)$)

The degree of a vertex is the number of edges incident with it, equivalently the cardinality of its neighbourhood in a simple graph.

The number of edges of a directed graph directed away from a given vertex. Assigning directions to undirected edges is an orientation of a graph.

The average vertex degree of a graph is the sum of its vertex degrees divided by its order. Double counting edge endpoints gives $\overline{d}(G)=2e(G)/|V(G)|$. It lies between the minimum degree of a graph and the maximum degree of a graph.

A graph is $d$-regular when every vertex has degree $d$.

The graph whose vertices are two-element subsets of a five-element set, with two vertices adjacent when the subsets are disjoint. It has ten vertices, degree three, zero common neighbours for adjacent pairs and one for nonadjacent pairs. Its adjacency matrix of a graph satisfies $A^2+A-2I=J$ and has eigenvalues $3,1,-2$ of multiplicities $1,5,4$.

If $A$ is the adjacency matrix of a $d$-regular graph and $\mathbf{e}=(1,\dots ,1{)}^T$, every row sum of $A$ is $d$, soIf the graph is connected, the $d$-eigenspace is exactly $\mathrm{span}\{\mathbf{e}\}$.

A connected regular graph has exactly two distinct adjacency eigenvalues if and only if it is a complete graph $K_n$ with $n\ge 2$. Its eigenvalues are $n-1$ on the constant vectors and $-1$ on their orthogonal complement.

In a finite graph with at least one edge, the degree-biased vertex distribution assigns probability proportional to the degree of a vertex. It is the distribution of an endpoint on a specified side of a uniformly random edge.

A universal vertex is adjacent to every other vertex of its graph.

The minimum degree $\delta (G)$ is the least degree of a vertex of $G$.

The maximum degree $\Delta (G)$ is the greatest degree of a vertex of $G$.

A graph is locally finite when every vertex has finite degree.

An exhaustion of a countable graph is an increasing sequence of finite connected graph vertex sets whose union is the whole graph vertex set. In a locally finite graph, every prescribed finite graph path together with all graph neighbours of its visited graph vertices lies in a sufficiently large exhaustion set. This permits coupling finite-volume and infinite-volume simple random walks up to a finite stopping time.